How do I prove that $\mathcal{H}$, with $\mathcal{VC}$ dimension $d$, shatters all subsets with size less than $d-1$?
We can show that it is not true by a counterexample. For example, $X = \{1,2,3\}$ and $\mathcal H = \{\{\},\{1\},\{2\},\{1,2\}\}$ is the finite set hypothesis class. By definition, in this case, the $\mathcal{VC}$ dimension of $\mathcal H$ over the domain $X$ is $d=2$. Although $A = \{3\} \subset X$, whose size is smaller than the $\mathcal{VC}$ dimension, i.e $|A|&lt;d=2$, it is not shattered by $\mathcal H$.